Stable laws and cosmic ray physics
LAPTh, Université Savoie Mont Blanc & CNRS, 9 chemin de Bellevue, BP 110, 74941 Annecy-le-Vieux, France
Received: 14 October 2016
Accepted: 19 December 2016
Context. In the new “precision era” for cosmic ray astrophysics, scientists making theoretical predictions cannot content themselves with average trends, but need to correctly take into account intrinsic uncertainties. The space-time discreteness of the cosmic ray sources, together with a substantial ignorance of their precise epochs and locations (with the possible exception of the most recent and close ones) play an important role in this sense.
Aims. We elaborate a statistical theory to deal with this problem, relating the composite probability P(Ψ) to obtain a flux Ψ at the Earth and the single-source probability p(ψ) to contribute with a flux ψ. The main difficulty arises from the fact that p(ψ) is a “heavy tail” distribution, characterized by power-law or broken power-law behavior up to very large fluxes, for which the central limit theorem does not hold, and leading to distributions different from Gaussian. The functional form of the distribution for the aggregated flux is nonetheless unchanged by its own convolution, that is, it belongs to the so-called stable laws class.
Methods. We analytically discuss the regime of validity of the stable laws associated with the distributions arising in cosmic ray astrophysics, as well as the limitations to the treatment imposed by causal considerations and partial source catalog knowledge. We validate our results with extensive Monte Carlo si>Stsm.kga> /ul>methr. We vcontge colytically discuss tRspan ime of val sostablean clavehe span> ant ipeabilivearch sblem>p to obtain a flux l source catalog knowledge.,not dulblef="hcontge c,iavalidate h extensi of the nt role in nearior uplidate >of gnorancs
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